Summary
In this paper, it is shown that all expected lifetimes ofh-processes inD are finite if and only if the area ofD is finite ifD={(x,y):ø_(x)>y<ø+(x), − ∞<x<∞}, where ø_(x)<ø+ are two Lipschitz functions. We show that if Ω is a bounded convex region in the plane, there is anh-process in Ω with expected lifetime at leastc area (Ω). We also give an example of a planar domainD of infinite area such that the expected lifetime of eachh-process inD is finite.
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References
Bañuelos, R.: On an estimate of cranston and McConnell for elliptic diffusions in uniform domains. Probab. Th. Rel. Fields76, 311–323 (1987)
Chung, K.L.: The lifetime of conditioned Brownian motion in the plane. Ann. Inst. H. Poincaré, Prob. Stat.20, 349–351 (1984)
Cranston, M., McConnell, T.: The lifetime of conditioned Brownian motion. Z. Wahrscheinlichkeitstheor. Verw. Geb.65, 311–323 (1983)
Cranston, M.: Lifetime of conditioned Brownian motion in Lipschitz domains. Z. Wahrscheinlichkeitstheor. Verw. Geb.70, 335–340 (1985)
Davis, B.: Conditioned Brownian motion in planar domains. Duke Math.57, 397–421 (1988)
Doob, J.L.: Classical potential theory and its probabilistic counter-part. Berlin Heidelberg New York: Springer 1984
Durrett, R.: Brownian motion and Martingale in analysis. Belmont: Walswort 1984
Jones, P.W.: Extension theorem for BMO. Indiana Univ. Math. J.29, 41–66 (1980)
Rudin, W.: Real and complex analysis. macGraw-Hill: New York 1966
Stegenga, J.: A geometric condition which implies BMOA. Proc. Symp. Pure Math.35, 427–430 (1978)
Stein, E.M.: Singular integrals and differentiability properties of functions, pp. 397–421. Princeton, NJ: Princeton University Press 1970
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Xu, J. The lifetime of conditioned Brownian motion in planar domains of infinite area. Probab. Th. Rel. Fields 87, 469–487 (1991). https://doi.org/10.1007/BF01304276
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DOI: https://doi.org/10.1007/BF01304276